9.2 Design of RFD Observer
167
By Schur complement, condition (9.24) can guarantee J 2 (T ) ≤ 0. This ends the
proof.
Therefore, the algorithm to design the RFD observer gain H i is concluded as:
(1) Acquire λ min , η max by solving LMI (9.19) and LMI (9.24), respectively.
(2) Assign λ 1 = λ min , η 1 = η max . If γ 1 , η 1 are solvable for LMI (9.19) and (9.24),
the optimal H i = P
−1
2
¯
H i can be gained. Or else, set η j = η j−1 + η, where
η > 0 is a constant that’s small enough and j ( j = 2, 3, . . .) is the jth iteration.
Repeat η j = η j−1 + η, until ¯
H i = ¯
H ji is solvable for LMIs (9.19) and (9.24).
Compute J 1e = λ 1 /η j . If η j is insolvable, we set λ l = λ l − 1 t , where λ > 0
is a constant that’s small enough and l(l = 2, 3, . . .) is the lth iteration. Repeat
λ l = λ l−1 + λ, until ¯
H i = ¯
H li is solvable for LMIs (9.19) and (9.24). Compute
J 2e = λ l η j .
(3) Combing with J min = min {J 1e , J 2e }, the RFD observer can be achieved. Note
the corresponding matrices P 1i , P 2i , H i = P
−1
2i
¯
H i and scalars λ, η.
9.3 Numeral Example
Consider the nonlinear multi-model jumping system (9.1) with parameters represented as:
A i =
1.8λ(i) 3.5
−5.0 −4λ(i)
, A hi =
−0.4λ(i)
0.6
−0.3 −0.4λ(i)
, B di =
0.4τ (i)
0.3
, B f i =
0.2
0.3τ (i)
,
C i =
0.3 0.3τ (i)
, C hi = [0.1τ (i) 0.1], C hi = [0.3τ (i) 0.2], D di = 0.4τ (i),
D f i = 1.5τ (i), M 1i =
0
0.5
, M 2i = 0.3, N 1i =
0.3 0.1
, N 2i =
0.3 0.2
, N 3i =
0.2, N 4i = 0.3, f i (x) =
0
cos (γ(i)x 1 (t))
, i = {1, 2, 3}.
{λ(i), τ (i), γ(i)} are the mode-dependent parameters taking values in sets {1.4,
1.3, 0.8} i=1 , {1.9, 1.8, 2.1} i=2 and {2.1, 1.4, 2.6} i=3 .
Besides, the transition rate matrix is given by:
=
⎡
⎣
−4 1.3 2.7
0.8 −1.5 0.7
0.7 0.8 −1.5
⎤
⎦ .
In this example, we provide three identical single hidden layer neural networks
with time delay h = 1 which can be used to approach three nonlinear functions
sin(γ(i)x 1 (t)) for each mode, and three LDIs are gained as {A 11 , A 12 , A 13 , A 14 },
{A 21 , A 22 , A 23 , A 24 }, {A 31 , A 32 , A 33 , A 34 }. The upper boundedness of approximation errors are ρ 1 = 0.2, ρ 2 = 0.19 and ρ 3 = 0.42, respectively. By solving LMIs
167
By Schur complement, condition (9.24) can guarantee J 2 (T ) ≤ 0. This ends the
proof.
Therefore, the algorithm to design the RFD observer gain H i is concluded as:
(1) Acquire λ min , η max by solving LMI (9.19) and LMI (9.24), respectively.
(2) Assign λ 1 = λ min , η 1 = η max . If γ 1 , η 1 are solvable for LMI (9.19) and (9.24),
the optimal H i = P
−1
2
¯
H i can be gained. Or else, set η j = η j−1 + η, where
η > 0 is a constant that’s small enough and j ( j = 2, 3, . . .) is the jth iteration.
Repeat η j = η j−1 + η, until ¯
H i = ¯
H ji is solvable for LMIs (9.19) and (9.24).
Compute J 1e = λ 1 /η j . If η j is insolvable, we set λ l = λ l − 1 t , where λ > 0
is a constant that’s small enough and l(l = 2, 3, . . .) is the lth iteration. Repeat
λ l = λ l−1 + λ, until ¯
H i = ¯
H li is solvable for LMIs (9.19) and (9.24). Compute
J 2e = λ l η j .
(3) Combing with J min = min {J 1e , J 2e }, the RFD observer can be achieved. Note
the corresponding matrices P 1i , P 2i , H i = P
−1
2i
¯
H i and scalars λ, η.
9.3 Numeral Example
Consider the nonlinear multi-model jumping system (9.1) with parameters represented as:
A i =
1.8λ(i) 3.5
−5.0 −4λ(i)
, A hi =
−0.4λ(i)
0.6
−0.3 −0.4λ(i)
, B di =
0.4τ (i)
0.3
, B f i =
0.2
0.3τ (i)
,
C i =
0.3 0.3τ (i)
, C hi = [0.1τ (i) 0.1], C hi = [0.3τ (i) 0.2], D di = 0.4τ (i),
D f i = 1.5τ (i), M 1i =
0
0.5
, M 2i = 0.3, N 1i =
0.3 0.1
, N 2i =
0.3 0.2
, N 3i =
0.2, N 4i = 0.3, f i (x) =
0
cos (γ(i)x 1 (t))
, i = {1, 2, 3}.
{λ(i), τ (i), γ(i)} are the mode-dependent parameters taking values in sets {1.4,
1.3, 0.8} i=1 , {1.9, 1.8, 2.1} i=2 and {2.1, 1.4, 2.6} i=3 .
Besides, the transition rate matrix is given by:
=
⎡
⎣
−4 1.3 2.7
0.8 −1.5 0.7
0.7 0.8 −1.5
⎤
⎦ .
In this example, we provide three identical single hidden layer neural networks
with time delay h = 1 which can be used to approach three nonlinear functions
sin(γ(i)x 1 (t)) for each mode, and three LDIs are gained as {A 11 , A 12 , A 13 , A 14 },
{A 21 , A 22 , A 23 , A 24 }, {A 31 , A 32 , A 33 , A 34 }. The upper boundedness of approximation errors are ρ 1 = 0.2, ρ 2 = 0.19 and ρ 3 = 0.42, respectively. By solving LMIs